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Condense each expression to a single logarithm.

2log_(6)u-8log_(6)v

Condense each expression to a single logarithm. $\newline$ $1$ ) $2 \log _{6} u-8 \log _{6} v$

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Product property of logarithms

Full solution

Q. Condense each expression to a single logarithm.

\newline

1

)

2 \log _{6} u-8 \log _{6} v

Identify Properties: Identify the properties of logarithms that can be used to condense the expression. $\newline$ The expression involves coefficients in front of the logarithms, which suggests the use of the power property of logarithms to move the coefficients inside the logarithms as exponents. $\newline$ Power Property: $a\log_b(P) = \log_b(P^a)$
Apply Power Property: Apply the power property to move the coefficients inside the logarithms as exponents. $\newline$ Using the power property, we can rewrite the expression as follows: $\newline$ $2\log_{6}u = \log_{6}(u^2)$ $\newline$ $8\log_{6}v = \log_{6}(v^8)$
Combine Using Quotient Property: Combine the two logarithms into a single logarithm using the quotient property. $\newline$ Quotient Property: $\log_b(P) - \log_b(Q) = \log_b\left(\frac{P}{Q}\right)$ $\newline$ So, $\log_{6}(u^2) - \log_{6}(v^8) = \log_{6}\left(\frac{u^2}{v^8}\right)$
Write Final Answer: Write the final answer as a single logarithm. $\newline$ The expression is now condensed to a single logarithm: $\newline$ $\log_{6}\left(\frac{u^2}{v^8}\right)$

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